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Jun
200
4
DarkSUSY: Computing Supersymmetric Dark Matter PropertiesNumerically
P. Gondolo1
Department of Physics, University of Utah,
115 South 1400 East, Suite 201, Salt Lake City, UT 84112-0830, USA
J. Edsjo2
Department of Physics, Stockholm University,
AlbaNova University Center, SE-106 91 Stockholm, Sweden
P. Ullio3
SISSA, via Beirut 4, 34014 Trieste, Italy
L. Bergstrom4
Department of Physics, Stockholm University,
AlbaNova University Center, SE-106 91 Stockholm, Sweden
M. Schelke5
Department of Physics, Stockholm University,
AlbaNova University Center, SE-106 91 Stockholm, Sweden
E.A. Baltz6
KIPAC, Stanford University
P.O. Box 90450,MS 29 Stanford, CA 94309, USA
Abstract
The question of the nature of the dark matter in the Universe remains one of the
most outstanding unsolved problems in basic science. One of the best motivated particle
physics candidates is the lightest supersymmetric particle, assumed to be the lightest
neutralino - a linear combination of the supersymmetric partners of the photon, the Z bo-
son and neutral scalar Higgs particles. Here we describe DarkSUSY, a publicly-available
advanced numerical package for neutralino dark matter calculations. In DarkSUSY one
can compute the neutralino density in the Universe today using precision methods which
include resonances, pair production thresholds and coannihilations. Masses and mix-
ings of supersymmetric particles can be computed within DarkSUSY or with the help of
external programs such as FeynHiggs, ISASUGRA and SUSPECT. Accelerator bounds can
be checked to identify viable dark matter candidates. DarkSUSY also computes a large
variety of astrophysical signals from neutralino dark matter, such as direct detection in
low-background counting experiments and indirect detection through antiprotons, an-
tideuterons, gamma-rays and positrons from the Galactic halo or high-energy neutrinos
from the center of the Earth or of the Sun. Here we describe the physics behind the
package. A detailed manual will be provided with the computer package.
Keywords: Supersymmetry; Dark Matter; Cosmology; Neutrino Telescopes; Gamma-ray Telescopes;
Cosmic Antiprotons; Cosmic Antideuterons; Cosmic Positrons; Direct Detection; Indirect Detection;
Numerical Code; Galactic Halo
[email protected] by the Swedish Research Council (VR), [email protected]@sissa.it4Partially supported by the Swedish Research Council (VR), [email protected]@[email protected]
1
Contents
1 Introduction 3
2 Definition of the Supersymmetric model 4
2.1 Neutralino and chargino sectors . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Sfermion masses and mixings . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Higgs sector and interface to FeynHiggs . . . . . . . . . . . . . . . . . . . . . 72.4 Interface to the mSUGRA codes ISASUGRA and SUSPECT . . . . . . . . . 7
3 Experimental constraints 7
4 Calculation of relic density 8
4.1 The Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.2 Thermal averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.3 Annihilation cross sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.4 A note about degrees of freedom and the annihilation rate . . . . . . . . . . . 11
5 Halo models 12
6 Detection rates 13
6.1 Direct detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136.2 Indirect detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
6.2.1 Monte Carlo simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 156.3 Neutrinos from the Sun and Earth . . . . . . . . . . . . . . . . . . . . . . . . 166.4 Indirect searches through antimatter signals . . . . . . . . . . . . . . . . . . . 176.5 Gamma rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
6.5.1 Sources and fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.5.2 Gamma rays with continuum energy spectrum . . . . . . . . . . . . . 206.5.3 χχ→ γγ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206.5.4 χχ→ Zγ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
6.6 Neutrinos from the halo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
7 Examples of results obtained with DarkSUSY 22
7.1 Benchmark models in mSUGRA . . . . . . . . . . . . . . . . . . . . . . . . . 227.2 Benchmark models in MSSM . . . . . . . . . . . . . . . . . . . . . . . . . . . 247.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
8 Conclusions 26
A Included coannihilations 26
2
1 Introduction
During the past few years, remarkable progress has been made in cosmology, both observa-tionally and theoretically. One of the outcomes of these rapid developments is the increasedconfidence that most of the mass of the observable Universe is of an unusual form, i.e., notmade up of ordinary baryonic matter. Recent analyses combining high-redshift supernovaluminosity distances, microwave background fluctuations and the dynamics and baryon frac-tion of galaxy clusters indicate that the present mass density of matter in the Universe ΩM =ρM/ρcrit normalized to the critical density ρcrit = 3H2
0/(8πGN ) = h2 × 1.9 · 10−29 g cm−3
is 0.1 ∼< ΩMh2∼< 0.2, which is considerable higher than the value ΩBh
2∼< 0.023 allowed by
big bang nucleosynthesis [1]. Here h ≃ 0.7± 0.15 is the present value of the Hubble constantin units of 100 km s−1 Mpc−1. A recent addition to the wealth of experimental data whichsupport the existence of non-baryonic dark matter is the WMAP microwave background re-sults [2]. In a joint analysis of the WMAP data together with other CMBR experiments,large-scale structure data, supernova data and the HST Key Project, the WMAP team report[2] ΩBh
2 = 0.0224 ± 0.009, ΩMh2 = 0.135 ± 0.009, and ΩΛ = 0.73 ± 0.04. Subtracting the
baryonic contribution ΩBh2 from the matter density ΩMh
2 leaves a non-baryonic cold darkmatter density ΩCDMh
2 = 0.113± 0.009.Also from the point of view of structure formation, non-baryonic dark matter seems to
be necessary, and the main part of it should consist of particles that were non-relativistic atthe time when structure formed (cold dark matter, CDM), thus excluding light neutrinos.Under reasonable assumptions, the WMAP collaboration, using also galaxy survey and Ly-αforest data, limit the contribution of neutrinos to Ωνh
2 < 0.0076 (95 % c.l.).A well-motivated particle physics candidate which has the required properties is the light-
est supersymmetric particle, assumed to be a neutralino [3, 4, 5]. (For thorough reviews ofsupersymmetric dark matter, see [6, 7].) Although supersymmetry is generally accepted as avery promising enlargement of the Standard Model of particle physics (for instance it wouldsolve the so-called hierarchy problem which essentially consists of understanding why theelectroweak scale is protected against Planck-scale corrections), little is known about whata realistic supersymmetric model would look like in its details. Therefore, it is a generalpractice to use the simplest possible model, the minimal supersymmetric enlargement of theStandard Model (the MSSM), usually with some additional simplifying assumptions. Ofcourse, there is no compelling reason why the actual model, if nature is supersymmetric atall, should be of this simplest kind. However, the MSSM serves as a useful template withwhich to test current ideas about detection, both in particle physics accelerators and in darkmatter experiments and contains many features which are expected to be universal for anysupersymmetric dark matter model. In fact, the knowledge gained by studying the MSSM indetail may be of even more general use, since it provides one specific example of a WIMP, aweakly interacting massive particle, which is generically what successful particle dark mattermodels require.7
Over several years, we have developed analytical and numerical tools for dealing withthe sometimes quite complex calculations necessary to go from given input parameters inthe MSSM to actual quantitative predictions of the relic density of the neutralinos in theUniverse, and the direct and indirect detection rates. The program package, which we havenamed DarkSUSY, has now reached a high level of sophistication and maturity, and we havereleased it publicly for the benefit of the scientific community working with problems relatedto dark matter. This paper describes the basic structure and the underlying physical andastrophysical formulas contained in DarkSUSY, as well as examples of its use. The version of
7A completely different type of particle is the axion [8], which is very light, but was never in thermalequilibrium. Its phenomenology is very different and will not be treated here.
3
the package described in this paper is DarkSUSY 4.1.For download of the latest version of DarkSUSY and for a more technical manual, please
visit the official DarkSUSY website, http://www.physto.se/~edsjo/darksusy/.
2 Definition of the Supersymmetric model
We work in the framework of the minimal supersymmetric extension of the Standard Modeldefined by, besides the particle content and gauge couplings required by supersymmetry, thesuperpotential (the notation used is that of [9] which marked the beginning of the developmentof DarkSUSY, and is similar to [10])
W = ǫij
(
−e∗RYE liLH
j1 − d∗
RYDqiLH
j1 + u∗
RYU qiLH
j2 − µHi
1Hj2
)
(1)
and the soft supersymmetry-breaking potential
Vsoft = ǫij
(
−e∗RAEYE liLH
j1 − d∗
RADYDqiLH
j1 + u∗
RAUYU qiLH
j2 −BµHi
1Hj2 + h.c.
)
+Hi∗1 m
21H
i1 +Hi∗
2 m22H
i2
+qi∗LM2
QqiL + li∗LM2
LliL + u∗
RM2U uR + d∗
RM2DdR + e∗RM
2E eR
+1
2M1BB +
1
2M2
(
W 3W 3 + 2W+W−)
+1
2M3gg. (2)
We give these and the following expressions since they contain our sign conventions. It shouldbe noted that various authors use various sign conventions, and many errors, often difficult tofind, can be avoided by keeping careful track of the signs, as we have tried to do consistentlyin DarkSUSY. Here i and j are SU(2) indices (ǫ12 = +1). The Yukawa couplings Y, thesoft trilinear couplings A and the soft sfermion masses M are 3 × 3 matrices in generationspace. e, l, u, d and q are the superfields of the leptons and sleptons and of the quarks andsquarks. A tilde indicates their respective scalar components. The L and R subscripts onthe sfermion fields refer to the chirality of their fermionic superpartners. B, W 3 and W± arethe fermionic superpartners of the U(1) and SU(2) gauge fields and g is the gluino field. µ isthe Higgsino mass parameter, M1, M2 and M3 are the gaugino mass parameters, B is a softbilinear coupling, while m2
1 and m22 are Higgs mass parameters.
These input parameters are contained in common blocks in the program. The full set ofinput parameters in version 4.1 of DarkSUSY, to be given at the weak scale, is mA, tanβ,µ, M1, M2, M3, AEaa, AUaa, ADaa, M
2Qaa, M
2Laa, M
2Uaa, M
2Daa, M
2Eaa (with a = 1, 2, 3).
The user may either provide these parameters directly to DarkSUSY or take advantage ofthe implementation of a MSSM pre-defined through a reduced number of parameters. In thismodel, the basic set of parameters is µ, M2, mA, tanβ, m0, At, Ab. Here mA is the massof the CP-odd Higgs boson and tanβ denotes the ratio, v2/v1, of the vacuum expectationvalues of the two neutral components of the SU(2) Higgs doublets. The parameters m0, At
and Ab are defined through the simplifying Ansatz: MQ = MU = MD = ME = ML = m01,AU = diag(0, 0, At), AD = diag(0, 0, Ab), AE = 0.
Below we will give some details to clarify our convention and additional features. Rele-vant quantities for phenomenological studies, such as the particle masses and mixings, areconsistently computed by DarkSUSY and available in arrays. The supersymmetry part of theprogram can thus be used for many applications, in particular for accelerator-based physicsstudies. Particle decay widths are also available, but currently only the widths of the Higgsbosons are calculated, the other particles having fictitious widths of 1 or 5 GeV (for the solepurpose of regularizing annihilation amplitudes close to poles).
4
2.1 Neutralino and chargino sectors
The neutralinos χ0i are linear combinations of the superpartners of the neutral gauge bosons,
B, W3 and of the neutral Higgsinos, H01 , H
02 . In this basis, their mass matrix is given by
Mχ01,2,3,4
=
M1 0 − g′v1√2
+ g′v2√2
0 M2 + gv1√2
− gv2√2
− g′v1√2
+ gv1√2
δ33 −µ+ g′v2√
2− gv2√
2−µ δ44
(3)
where g and g′ are the gauge coupling constants of SU(2) and U(1). δ33 and δ44 are themost important one-loop corrections. These can change the neutralino masses by a few GeVup or down and are only important when there is a severe mass degeneracy of the lightestneutralinos and/or charginos. The expressions for δ33 and δ44 used in DarkSUSY are takenfrom [11, 12] (the tree-level values can optionally be chosen).
The neutralino mass matrix, Eq. (3), is diagonalized analytically and evaluated numeri-cally to give the masses and compositions of four neutral Majorana states,
χ0i = Ni1B +Ni2W
3 +Ni3H01 +Ni4H
02 , (4)
the lightest of which, χ01 to be called χ for simplicity, is then the candidate for the particle
making up the dark matter in the Universe. The neutralinos are ordered in mass such thatmχ0
1< mχ0
2< mχ0
3< mχ0
4and the eigenvalues are real with a complex N .
The charginos are linear combinations of the charged gauge bosons W± and of the chargedHiggsinos H−
1 , H+2 . Their mass terms are given by
( W− H−1 ) Mχ±
(
W+
H+2
)
+ h.c. (5)
Their mass matrix,
Mχ± =
(
M2 gv2gv1 µ
)
, (6)
is diagonalized by the following linear combinations
χ−i = Ui1W
− + Ui2H−1 , (7)
χ+i = Vi1W
+ + Vi2H+2 . (8)
We choose det(U) = 1 and U∗Mχ±V † = diag(mχ±1,mχ±
2) with non-negative chargino masses
mχ±i≥ 0. We do not include any one-loop corrections to the chargino masses since they are
negligible compared to the corrections δ33 and δ44 introduced above for the neutralino masses[11].
2.2 Sfermion masses and mixings
When discussing the squark mass matrix including mixing, it is convenient to choose a basiswhere the squarks are rotated in the same way as the corresponding quarks in the StandardModel. We follow the conventions of the Particle Data Group [13] and put the mixing in theleft-handed d-quark fields, so that the definition of the Cabibbo-Kobayashi-Maskawa matrixis K = V1V
†2, where V1 (V2) rotates the interaction left-handed u-quark (d-quark) fields
5
to mass eigenstates. For sleptons we choose an analogous basis, but since in DarkSUSY 4.1neutrinos are assumed to be massless, no analog of the CKM matrix appears.
The general 6× 6 u- and d-squark mass matrices are
M2u =
(
M2Q +m†
umu +DuLL1 m†
u(A†U − µ∗ cotβ)
(AU − µ cotβ)mu M2U +mum
†u +Du
RR1
)
, (9)
M2d=
(
K†M2QK+mdm
†d +Dd
LL1 m†d(A
†D − µ∗ tanβ)
(AD − µ tanβ)md M2D +m
†dmd +Dd
RR1
)
, (10)
The general sneutrino and charged slepton mass matrices are (for massless neutrinos)
M2ν = M2
L +DνLL1 (11)
M2e =
(
M2L +mem
†e +De
LL1 m†e(A
†E − µ∗ tanβ)
(AE − µ tanβ)me M2E +m†
eme +DeRR1
)
. (12)
HereDf
LL = m2Z cos 2β(T3f − ef sin
2 θW ), (13)
DfRR = m2
Z cos(2β)ef sin2 θW (14)
where T3f is the third component of the weak isospin and ef is the charge in units of theabsolute value of the electron charge, e. In the chosen basis, we havemu = diag (mu,mc,mt),md = diag (md,ms,mb) and me = diag(me,mµ,mτ ).
The slepton and squark mass eigenstates fk (νk with k = 1, 2, 3 and ek, uk and dk withk = 1, . . . , 6) diagonalize the previous mass matrices and are related to the current sfermioneigenstates fLa and fRa (a = 1, 2, 3) via
fLa =
6∑
k=1
fkΓ∗kaFL , (15)
fRa =
6∑
k=1
fkΓ∗kaFR . (16)
The squark and charged slepton mixing matrices ΓUL, ΓUR, ΓDL, ΓDR, ΓEL, and ΓER havedimension 6× 3, while the sneutrino mixing matrix ΓνL has dimension 3× 3.
The current version of DarkSUSY allows only for diagonal matrices AU , AD, AE , MQ,MU , MD, ME, and ML. This ansatz, while not being the most general, implies the absenceof tree-level flavor changing neutral currents in all sectors of the model. It also allows thesquark mass matrices to be diagonalized analytically. For example, for the top squark onehas, in terms of the top squark mixing angle θt,
Γt1 tUL = Γt2 t
UR = cos θt, Γt2 tUL = −Γt1 t
UR = sin θt. (17)
The sfermion masses are obtained with the diagonalization just described.To facilitate comparisons with the results of other authors, DarkSUSY allows for special
values of the sfermion masses to be set in the program. A common value can be assigned toall squark masses, and an independent common value to all slepton masses. Alternatively,to enforce the sfermions to be heavier than the lightest neutralino (which we want to bethe LSP), the squarks and sleptons masses can be set equal to the maximum between theneutralino mass and a specified value. In the special cases just described, no mixing isassumed between sfermions. It must be noted that the special choices described in thisparagraph are mathematically inconsistent within the MSSM, but are often made for thesake of simplicity.
6
2.3 Higgs sector and interface to FeynHiggs
The Higgs masses receive radiative corrections and DarkSUSY includes several options forcalculating these. The default in DarkSUSY is to use FeynHiggsFast [14] for the Higgs masscalculations. When higher accuracy is needed, it is possible to instead use the full FeynHiggs[15, 16] package.
2.4 Interface to the mSUGRA codes ISASUGRA and SUSPECT
Given the modular structure of DarkSUSY, the user may also run the package using as inputfor the MSSM definition the output, still at the low energy scale, from an external package.An example of usage under such mode is given in case of the minimal supergravity (mSUGRA)model: in the release, we provide an interface to the output of the ISASUGRA code, as includedin ISAJET [17] for the current ISASUGRA 7.69 version, as well as an an interface to SUSPECT
[18] which can be used as an alternative. The interfaces to DarkSUSY are at the level of thefull spectrum of masses and mixings, including, for consistency, those for the Higgs sector.Of importance for relic density calculations near a Higgs boson resonance is the possibilityof including supersymmetric corrections to the bottom, top, and tau Yukawa couplings assupplied by ISASUGRA.
3 Experimental constraints
Accelerator bounds can be checked by a call to a subroutine. By modifying an option, theuser can impose bounds as of different moments in time. The default option in version 4.1adopts the 2002 limits by the Particle Data Group [19] modified as described below. Theuser is also free to use his or her own routine to check for experimental bounds, in which casethere is only need to provide an interface to DarkSUSY.
For the theoretical prediction of the rare decay b→ sγ we have implemented the completenext-to-leading order (NLO) Standard Model calculation and the dominant NLO supersym-metric corrections. For the NLO QCD calculation of the Standard Model prediction wehave used the expressions of reference [20] into which we have inserted the updated so-called“Magic numbers” of [21]. Our implementation of the Standard Model calculation gives abranching ratio BR[B → Xs γ] = 3.72 × 10−4 for a photon energy greater than mb/20.This result agrees to within 1% with the result stated in [21], but is around 10% largerthan the result of previous analyses. The latter is due to the fact that in the reference
[20] they replaced mpolec /mpole
b by mMSc (µ)/mpole
b (with µ ∈ [mc,mb]) in the matrix element〈Xsγ | (sc)V −A(cb)V−A | b〉.
The supersymmetric correction to b → sγ has been divided into a contribution from atwo Higgs doublet model and a contribution from supersymmetric particles. The expressionsfor the NLO contributions in the two Higgs doublet model has been taken from [22]. Thecorrections due to supersymmetric particles are calculated under the assumption of minimalflavour violation. The dominant LO contributions which are valid even in the large tanβregime was taken from ref. [23], and we also followed their guideline on how the NLO QCDexpressions of [24] should be expanded to the large tanβ regime.
For the current experimental bound on b → sγ we take the value stated by the ParticleData Group 2002 [19]. This is an average between the CLEO and the Belle measurementsand amounts to BR[B → Xs γ] = (3.3±0.4)×10−4. To this we add a theoretical uncertaintywhich we set to ±0.5 × 10−4. The final constraint on the branching ratio then becomes2.0× 10−4 ≤ BR[B → Xs γ] ≤ 4.6× 10−4.
7
Recently, much interest has been given to the possible contribution of supersymmetryto (g − 2)µ. Although the discrepancy with the Standard Model result is now below 3σ,we include for convenience a calculation of the anomalous moment of the muon (g − 2)µ inDarkSUSY.
4 Calculation of relic density
The WMAP microwave background experiment [2], combined with other sets of data, givesa quite precise determination of the cold dark matter density ΩCDMh
2 = 0.113 ± 0.009.We would like DarkSUSY to compute the relic density of neutralinos to at least the sameprecision.
We use in DarkSUSY the full cross section and solve the Boltzmann equation numericallywith the method given in [25, 26]. In this way we automatically take care of thresholds andresonances.
When any other supersymmetric particles are close in mass to the lightest neutralinothey will also be present at the time when the neutralino freezes out in the early Universe.When this happens so-called coannihilations can take place between all these supersymmetricparticles present at freeze-out. Coannihilations were first pointed out by Binetruy, Girardi,and Salati [27] in a non-supersymmetric model with several Higgs bosons. Griest and Seckel[28] investigated them in the MSSM for the case where squarks are of about the same mass asthe lightest neutralino. Later, coannihilations between the lightest neutralino and the lightestchargino were investigated in [29, 30, 11]. Several authors have also included coannihilationswith sfermions[31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46]. In DarkSUSYwe have implemented all coannihilations between the neutralinos, charginos and sfermionsas calculated in [46].
Compared to other recent calculations, we believe ours is the most precise calculationavailable at present. The standard lore so far has been to calculate the thermal average ofthe annihilation cross section by expanding to first power in temperature over mass and im-plementing an approximate solution to the evolution equation which estimates the freeze outtemperature without fully solving the equation (see, e.g., Kolb and Turner [47]). Sometimesthis is refined by including resonances and threshold corrections [28]. Among recent studies,this approach is taken in e.g. Refs. [36, 37]. Other refinements include, e.g., solving thedensity evolution equation numerically but still using an approximation to thermal effects inthe cross section [31, 32, 33, 34, 35], or calculating the thermal average precisely but usingan approximate solution to the density equation [38, 39, 40]. At the same time, only in a fewof the quoted papers the full set of initial states has been included. As already mentioned,the present calculation includes all initial states, performs an accurate thermal average andgives a very precise solution to the evolution equation. Though the inclusion of initial statesfermions in the DarkSUSY package is a new feature introduced in the present work, othergroups [41, 42, 43] have earlier introduced some sfermion coannihilations in an interface withthe old DarkSUSY version.
To gain calculational speed we only include the particles with masses below fcomχ. Themass fraction parameter fco is by default set to 2.1 or 1.4 depending on how the relic densityroutines are called (very high precision or fast calculation), but can be set to any value bythe user.
4.1 The Boltzmann equation
We will here outline the procedure developed in [26] which is used in DarkSUSY. For moredetails, see [26].
8
Consider annihilation of N supersymmetric particles with masses mi and internal degreesof freedom gi. Order them such that m1 ≤ m2 ≤ · · · ≤ mN−1 ≤ mN . For the lightestneutralino, the notation m1 and mχ will be used interchangeably.
Since we assume that R-parity holds, all supersymmetric particles will eventually decayto the LSP and we thus only have to consider the total number density of supersymmetricparticles n =
∑Ni=1 ni. Furthermore, the scattering rate of supersymmetric particles off
particles in the thermal background is much faster than their annihilation rate, becausethe scattering cross sections σ′
Xij are of the same order of magnitude as the annihilationcross sections σij but the background particle density nX is much larger than each of thesupersymmetric particle densities ni when the former are relativistic and the latter are non-relativistic, and so suppressed by a Boltzmann factor [48]. Hence, the χi distributions remainin kinetic thermal equilibrium during their freeze-out. Combining these effects, we arriveat the following Boltzmann equation for the summed number density of supersymmetricparticles
dn
dt= −3Hn− 〈σeffv〉
(
n2 − n2eq
)
(18)
where
〈σeffv〉 =∑
ij
〈σijvij〉neqi
neq
neqj
neq. (19)
with
vij =
√
(pi · pj)2 −m2im
2j
EiEj. (20)
4.2 Thermal averaging
Using the Maxwell-Boltzmann approximation for the velocity distributions one can derivethe following expression for the thermally averaged annihilation cross section [26]
〈σeffv〉 =∫∞0 dpeffp
2effWeffK1
(√s
T
)
m41T[
∑
igig1
m2i
m21K2
(
mi
T
)
]2 . (21)
where K1 (K2) is the modified Bessel function of the second kind of order 1 (2), T is thetemperature, s is one of the Mandelstam variables and
Weff =∑
ij
pijpeff
gigjg21
Wij
=∑
ij
√
[s− (mi −mj)2][s− (mi +mj)2]
s(s− 4m21)
gigjg21
Wij . (22)
In this equation, we have defined the momentum
pij =
[
s− (mi +mj)2]1/2 [
s− (mi −mj)2]1/2
2√s
, (23)
the invariant annihilation rate
Wij = 4pij√sσij = 4σij
√
(pi · pj)2 −m2im
2j = 4EiEjσijvij (24)
9
and the effective momentum
peff = p11 =1
2
√
s− 4m21. (25)
Since Wij(s) = 0 for s ≤ (mi + mj)2, the radicand in Eq. (22) is never negative. For a
two-body final state, Wij is given by
W 2−bodyij =
|k|16π2gigjSf
√s
∑
internal d.o.f.
∫
|M|2 dΩ, (26)
where k is the final center-of-mass momentum, Sf is a symmetry factor equal to 2 for identicalfinal particles, and the integration is over all possible outgoing directions of one of the finalparticles. As usual, an average over initial internal degrees of freedom is performed.
4.3 Annihilation cross sections
In DarkSUSY, all two-body final state cross sections at tree level are computed for all coan-nihilations between neutralinos, charginos and sfermions. A complete list is given in table 3in Appendix A.
The calculation of the relic density is, due to its complexity, the most time-consumingtask of DarkSUSY. For the neutralino-neutralino, chargino-neutralino and chargino-charginoinitial states, to achieve efficient numerical computation, contributing diagrams have beenclassified according to their topology (s-, t- or u-channel) and to the spin of the particlesinvolved. The helicity amplitudes for each type of diagram have been computed analyticallywith Reduce [49] using general expressions for the vertex couplings. The Reduce output hasbeen automatically converted to FORTRAN subroutines called by DarkSUSY.
The strength of the helicity amplitude method is that the analytical calculation of a givendiagram only has to be performed once and the summing of the contributing diagrams foreach given set of initial and final states can be done numerically afterwards.
The Feynman diagrams are summed numerically for each annihilation channel ij → kl.We then sum the squares of the helicity amplitudes and sum the contributions of all annihi-lation channels. Explicitly, DarkSUSY computes
dWeff
d cos θ=∑
ijkl
pijpkl32πSkl
√s
∑
helicities
∣
∣
∣
∣
∣
∣
∑
diagrams
M(ij → kl)
∣
∣
∣
∣
∣
∣
2
(27)
where θ is the angle between particles k and i. Finally, a numerical integration over cos θ isperformed by means of an adaptive method [50].
In rare cases, there can be resonances in the t- or u-channels. This can happen whenthe masses of the initial state particles lie between the masses of the final state particles.At certain values of cos θ, the momentum transfer is time-like and matches the mass of theexchanged particle. We regulate the divergence by assigning a small width of 5 GeV to theneutralinos and charginos and 1 GeV to the sfermions. The results are not sensitive to theexact choice of this width.
For the coannihilation diagrams with sfermions, the calculations are done with Form [51]and automatically converted into FORTRAN subroutines.
In the relic density routines, the calculation of the effective invariant rateWeff is the mosttime-consuming part. Fortunately, thanks to the remarkable feature of Eq. (21), Weff(peff)does not depend on the temperature T , and it can be tabulated once for each model.
To perform the thermal average in Eq. (21), we integrate over peff by means of adaptivegaussian integration, using a spline routine to interpolate in the (peff ,Weff) table. To avoid
10
numerical problems in the integration routine or in the spline routine, we split the integrationinterval at each sharp threshold. We also explicitly check for each MSSM model that thespline routine behaves well close to thresholds and resonances.
We then integrate the density evolution equation, Eq. (18). For numerical reasons, wedo not integrate the equation directly, but instead rephrase it as an evolution equation forthe abundance, Y = n/s (with s being the entropy density) and use x = mχ/T as ourintegration variable instead of time (see [26] for details). The numerical integration is subtlesince the equation is “stiff.” For this purpose, we developed an implicit trapezoidal methodwith adaptive stepsize. In short, if the equation for Y is written as dY/dx = λ(Y 2 − Y 2
e ),with Ye being the equilibrium value at temperature T , the numerical integration is based onthe recurrence
Yi+1 =Ci
1 +√
1 + hλi+1Ci
(28)
where h is the stepsize and Ci = 2Yi + h[λi+1Y2e,i+1 + λi(Y
2e,i − Y 2
i )]. The stepsize isadapted (reduced) if |(Yi+1 − yi+1)/Yi+1| exceeds a given tolerance. Here yi+1 = (ci/2)(1 +√
1 + hλi+1ci)−1 with ci = 4(Yi + hλiY
2e,i).
There are some loop-induced final states, such as two gluons, two photons or a Z0 bosonand a photon which could in principle give a non-negligible contribution to the annihilationrate and lower the relic abundance somewhat. The cross sections for these processes arevery complicated already in the limit of zero velocity, and we therefore assume that theinvariant rateW for these one-loop processes are constant and equal to their zero-momentumexpressions. These processes can be excluded from the calculation by setting appropriateparameters in the code.
The relic density routines can be called in a precise way where all integrations are per-formed to a precision better than 1% and coannihilations are included up to a mass differenceof fco = 2.1. It can also be called in a faster way, where the precision of the integrations areof the order of 1% and coannihilations are included up to fco = 1.4. Usually, the differencebetween the precise and fast method is completely negligible, but in rare cases it can be of afew %. The fast option is considerably faster and should be sufficient for most purposes. Foradvanced users, it is also possible to manually decide exactly which coannihilation processesto include.
For users with less demand of calculational precision, we also provide in DarkSUSY theoption of a “quick-and-dirty” method of computing the relic density, essentially according tothe textbook treatment in [47]. It should be realised, however, that this may sometimes givea computed relic density which is wrong by orders of magnitude.
4.4 A note about degrees of freedom and the annihilation rate
We end this section with a comment on the internal degrees of freedom gi. A neutralinois a Majorana fermion (it is its own antiparticle) and has two internal degrees of freedom,gχ0
i= 2. A chargino can be treated either as two separate species χ+
i and χ−i , each with
internal degrees of freedom gχ+ = gχ− = 2, or, more simply, as a single species χ±i with
gχ±i= 4 internal degrees of freedom. We choose the latter convention in DarkSUSY and use
the analogous conventions for sfermions (see [46] for details).The counting of states in annihilation processes for Majorana fermions is non-trivial,
and has led to a factor-of-two ambiguity in the literature which also propagated into earlierversions of DarkSUSY. The current version of DarkSUSY contains the correct normalizationof annihilation rates, namely σv in the Boltzmann equation and σv/2 for annihilation in thehalo. The clearest way to see the origin of the factor of 1/2 is probably to go back to theBoltzmann equation, [52]. In essence, one can view σv as the thermal average (averaged over
11
momentum and angles) of the cross section times velocity in the zero momentum limit; in thisaverage one integrates over all possible angles. For identical particles in the initial state, oneincludes each possible initial state twice, therefore one needs to compensate by dividing by afactor of 2; the prefactor in the zero-momentum limit becomes then σv/2. In the Boltzmannequation describing the time evolution of the neutralino number density the 1/2 does notappear as it is compensated by the factor of 2 one has to include because 2 neutralinos aredepleted per annihilation, but we need to include the factor of 1/2 explicitly in other caseswhere we need the annihilation rate (like for annihilation in the halo). Another way to viewthis is to think of σ as to the annihilation cross section for a given pair of particles. Letthe number of neutralinos in a given volume be N ; the annihilation rate would be given byσv times the number of pairs, which is N(N − 1)/2. In the continuum limit this reduces toσvn2/2.
5 Halo models
The modelling of the distribution of dark matter particles in the Milky Way dark halo playsa major role in estimates of dark matter detection rates. On this issue, however, there isno well-established framework we can refer to. Available dynamical measurements, suchas, e.g., the mapping of the rotation curve, the local field of acceleration of stars in thedirection perpendicular to the disk, or the motion of the satellites in the outskirts of theGalaxy, provide some constraints on the dark matter density profile, but lack the precisionneeded to derive a refined model. The guideline for the future will be N-body simulations ofhierarchical clustering in cold dark matter cosmologies, which are starting to resolve the innerstructure of individual galactic halos. At present, however, the translation of such resultsinto the detailed model we need for the Milky Way still relies on large extrapolations, and,to some extent, faces the problem of possible discrepancies between some of its propertiesand observations in real galaxies (for a recent review see, e.g., [53]).
In light of these large uncertainties, the definition of the model for the dark matter haloin DarkSUSY has been kept in a very general format. Two sets of properties need to beimplemented:
a) Local properties: To derive counting rates in direct detection and χ-induced neutrinofluxes from the center of the Sun and the Earth, the user has to specify the halomass density ρ0 at our galactocentric distance R0 and the relative particle velocitydistribution f(u), where u is the modulus of the velocity ~u in the rest frame of the Sun(or Earth, respectively) and an average of incident angles has been assumed. Optionsto set f(u) include the possibility to implement the expression valid for a (truncated)isothermal sphere, or the numerical interpolation of a table of pairs (fi, ui) providedby the user. We have also implemented routines which compute the function f(u) inthe Earth or the Sun rest frame for a given isotropic velocity distribution function inthe Galactic reference frame as needed for direct detection and for neutralino capturerates by the Sun. Finally a numerical table for f(u) in the Earth frame and includingthe modelling of the diffusion process of neutralinos through the solar system [54] isavailable to compute capture rates by the Earth.
b) Global properties: To compute indirect signals from pair annihilations in the halo, thefull dark matter mass density profile is needed. Charged cosmic ray fluxes are com-puted in two dimensional models and a generic axisymmetric profile ρ(R, z) can beimplemented by the user. Line of sight integrals with angular averaging over accep-tance angles, needed to compute gamma-ray fluxes, are given for spherically symmetric
12
profiles ρ(r). The options to specify ρ include several analytic forms, with most classesof profiles proposed in recent years [55, 56, 57, 58], as well as the numerical interpola-tion of a table of pairs (ρi, ri) provided by the user. It is also possible to switch on anoption to compute the signals for annihilations taking place in a population of small,unresolved clumps. In this case the user should specify the probability distribution forthe clumps and the average enhancement of the source per unit volume compared tothe smooth halo scenario.
An option to rescale ρ0 and ρ(r) by the quantity Ωh2/Ωminh2, where Ωh2 is the neutralino
relic abundance and Ωminh2 a minimum reference value for neutralinos to provide most of
the dark matter in our Galaxy, is available for the case where subdominant dark mattercandidates are considered. The effect can be introduced a posteriori as well for all detectiontechniques except for neutrino fluxes, where the response to rescaling is non-linear.
A separate package interfaced to DarkSUSY with self consistent pairs (ρ(r), f(u)) inΛCDM inspired models which fit available dynamical constraints will be available shortlyfrom one of the authors [59].
6 Detection rates
For the detection rates of neutralino dark matter we have used the rates as calculated in Refs.[52, 60, 9, 61, 62, 63, 64, 65, 66], with some improvements which we report in this Sectionwhere the formulas used in DarkSUSY are presented.
6.1 Direct detection
The current version of DarkSUSY provides the neutralino-proton and neutralino-neutron scat-tering cross sections (spin-independent and spin-dependent), as well as cross sections and formfactors for elastic scattering of neutralinos off nuclei.
The rate for direct detection of galactic neutralinos can be written as
dR
dE=∑
i
ciρχσχi|Fi(qi)|2
2mχµ2iχ
∫
v>√
MiE/2µ2χi
f(v, t)
vd3v. (29)
The sum is over the nuclear species in the detector, ci being the detector mass fractionin nuclear species i. Mi is the nuclear mass, and µχi = mχMi/(mχ +Mi) is the reducedneutralino–nucleus mass. Moreover, ρχ is the local neutralino density, v the neutralinovelocity relative to the detector, v = |v|, and f(v, t)d3v the neutralino velocity distribution.Finally, σχi is the total scattering cross section of a WIMP off a fictitious point-like nucleus,|Fi(qi)|2 is a nuclear form factor that depends on the momentum transfer qi =
√2MiE and
is normalized to Fi(0) = 1. The integration is over all neutralino speeds that can impart arecoil energy E to the nucleus.
The cross section σχi scales differently for spin-dependent and spin-independent interac-tions. For spin-independent interactions with a nucleus with Z protons and A−Z neutrons,one has
σSIχi =
µ2χi
π
∣
∣ZGps + (A− Z)Gn
s
∣
∣
2, (30)
where Gps and Gn
s are the scalar four-fermion couplings of the WIMP with point-like protonsand neutrons, respectively (see, e.g., [67], and below). As default, the spin-independent formfactor in DarkSUSY is taken to be of the Helm form
|F SI(q)|2 =
(
3j1(qR1)
qR1
)2
e−q2s2 , (31)
13
with j1 a spherical Bessel function of first kind, R1 =√R2 − 5s2, R = [0.9(M/GeV)1/3 +
0.3] fm and s = 1 fm. An exponential form factor is also available as an option.For spin-dependent interactions, one has instead, at zero momentum transfer,
σSDχi =
4µ2χi
π
J + 1
J
∣
∣〈Sp〉Gpa + 〈Sn〉Gn
a
∣
∣
2, (32)
where J is the nuclear spin, 〈Sp〉 and 〈Sn〉 are the expectation values of the spin of theprotons and neutrons in the nucleus, respectively, and Gp
a and Gna are the axial four-fermion
couplings of the WIMP with point-like protons and neutrons [67, 68]. At finite momentumtransfer, the spin-dependent cross section times the form factor reads
σSDχi |F SD
i (q)|2 =4µ2
χi
2J + 1
[
(Gpa)
2Spp(q) + (Gna )
2Snn(q) +GpaG
naSpn(q)
]
, (33)
where Spp(q) = S00(q) + S11(q) + S01(q), Snn(q) = S00(q) + S11(q) − S01(q), and Spn(q) =2[S00(q)−S11(q)]. The spin structure functions S00(q), S11(q), and S01(q) are defined in [69]and are given in the literature.
For protons and neutrons, the previous expressions reduce to
σSIχp =
µ2χp
π|Gp
s |2, σSIχn =
µ2χn
π|Gn
s |2, σSDχp =
3µ2χp
π|Gp
a|2, σSDχn =
3µ2χn
π|Gn
a |2. (34)
For the neutralino, the scalar and axial four-fermion couplings with the proton and neu-tron arise from squark, Higgs and Z boson exchange. In DarkSUSY, the default expressionsfor a nucleon N = n, p are
GNs =
∑
q=u,d,s,c,b,t
〈N |qq|N〉
1
2
6∑
k=1
gLqkχqgRqkχq
m2qk
−∑
h=H1,H2
ghχχghqqm2
h
, (35)
GNa =
∑
q=u,d,s
(∆q)N
(
gZχχgZqq
m2Z
+1
8
6∑
k=1
g2Lqkχq+ g2Rqkχq
m2qk
)
, (36)
where gabc are the coupling constants in the vertex involving particles abc (see [9] and [65]for explicit expressions). The more complicated expressions of Drees and Nojiri [70] are alsoavailable as an option.
Default values of the parameters in DarkSUSY are [71, 72] (with 〈N |qq|N〉 = fNTqmN/mq)
fpTu = 0.023, fp
Td = 0.034, fpTs = 0.14, fp
Tc = fpTb = fp
Tt = 0.0595, (37)
fnTu = 0.019, fn
Td = 0.041, fnTs = 0.14, fn
Tc = fnTb = fn
Tt = 0.0592. (38)
(∆u)p = (∆d)n = 0.77, (∆d)p = (∆u)n = −0.40, (∆s)p = (∆s)n = −0.12. (39)
Other sets of values for these parameters are available. These values can also be overriddenby the user.
6.2 Indirect detection
There are several ways of detecting dark matter particles indirectly. Pair annihilations of darkmatter particles in the Galactic halo may give an exotic component in positron, antiproton orantideuteron cosmic-rays and gamma-rays. There may also be annihilation in astrophysicalenvironments where the density of neutralinos may be enhanced, such as annihilation in thecenter of the Earth or Sun (detected in neutrino telescopes through a high-energy neutrinoflux) or near the central black hole of the Galaxy.
14
6.2.1 Monte Carlo simulations
In several of the indirect detection processes below we need to evaluate the yield of differentparticles per neutralino annihilation. The hadronization and/or decay of the annihilationproducts are simulated with Pythia [73] 6.154 in essentially the same way (with a few dif-ferences) for all these processes and we here describe how the simulations are done. We candivide the simulations into two groups: a) annihilation in the Earth and the Sun and b)annihilation in the halo. In both cases the simulations are done for a set of 18 neutralinomasses, mχ = 10, 25, 50, 80.3, 91.2, 100, 150, 176, 200, 250, 350, 500, 750, 1000, 1500, 2000,3000 and 5000 GeV. We tabulate the yields and then interpolate these tables in DarkSUSY.
These two groups of simulations differ slightly in other aspects, namely
a) Annihilation in the Earth and the Sun. In this case we are mainly interested in the fluxof high energy muon neutrinos and neutrino-induced muons at a neutrino telescope.We simulate 6 ‘fundamental’ annihilation channels, cc, bb, tt, τ+τ−, W+W− and Z0Z0
(if kinematically allowed) for each mass listed above. The lighter leptons and quarks donot contribute significantly and can safely be neglected. Pions and kaons get stoppedbefore they decay and are thus made stable in the Pythia simulations so that they donot produce any neutrinos. For annihilation channels containing Higgs bosons, we cancalculate the yield from these fundamental channels by letting the Higgs bosons decayin flight (see below). We also take into account the energy losses of B-mesons in theSun and the Earth by following the approximate treatment of [74] but with updatedB-meson interaction cross sections as given in [65]. Neutrino-interactions on the wayout of the Sun are simulated with Pythia including neutral current interactions andcharged-current interactions as a neutrino-loss. The neutrino-nucleon charged currentinteractions close to the detector are also simulated with Pythia and finally the mul-tiple Coulomb scattering of the muon on its way to the detector is calculated usingdistributions from [13]. For more details on these simulations see [75, 76].
For each annihilation channel and mass we simulate 1.25× 106 annihilations and tabu-late the final results as a neutrino-yield, neutrino-to-muon conversion rate and a muonyield differential in energy and angle from the center of the Sun/Earth. We also tab-ulate the integrated yield above a given threshold and below an open-angle θ. We as-sumed throughout that the surrounding medium is water with a density of 1.0 g/cm3.Hence, the neutrino-to-muon conversion rates have to be multiplied by the density ofthe medium. In the muon fluxes, the density cancels out (to within a few percent). Forthe neutrino-nucleon cross sections, we have used the parameterizations in [65].
b) Annihilation in the halo. The simulations are here simpler since we do not have asurrounding medium that can stop the annihilation products. We here simulate for8 ‘fundamental’ annihilation channels cc, bb, tt, τ+τ−, W+W−, Z0Z0, gg and µ+µ−.Compared to the simulations in the Earth and the Sun, we now let pions and kaonsdecay and we also let antineutrons decay to antiprotons. For each mass we simulate2.5 × 106 annihilations and tabulate the yield of antiprotons, positrons, gamma rays(not the gamma lines), muon neutrinos and neutrino-to-muon conversion rates andthe neutrino-induced muon yield, where in the last two cases the neutrino-nucleoninteractions has been simulated with Pythia as outlined above.
With these simulations, we can calculate the yield for any of the above mentioned particlesfor a given MSSM model. For the Higgs bosons, which decay in flight, an integration overthe angle of the decay products with respect to the direction of the Higgs boson is performed.Given the branching ratios for different annihilation channels it is then straightforward to
15
compute the muon flux above any given energy threshold and within any angular regionaround the Sun or the center of the Earth.
6.3 Neutrinos from the Sun and Earth
One of the most promising indirect detection method [77] relies on the fact that scatteringof halo neutralinos by the Sun and the planets, in particular the Earth, during the severalbillion years that the Solar system has existed, will have trapped these neutralinos withinthese astrophysical bodies. Being trapped within the Solar or terrestrial material, theywill sink towards the center, where a considerable enrichment and corresponding increase ofannihilation rate will occur.
Searches for neutralino annihilation into neutrinos will be subject to extensive experi-mental investigations in view of the new neutrino telescopes (AMANDA, IceCube, Baikal,NESTOR, ANTARES) planned or under construction [78]. A high-energy neutrino signal inthe direction of the center of the Sun or Earth is an excellent experimental signature whichmay stand up against the background of neutrinos generated by cosmic-ray interactions inthe Earth’s atmosphere.
The detector energy threshold for “small” neutrino telescopes like Baksan, MACRO andSuper-Kamiokande is around 1 GeV. Large area neutrino telescopes in the ocean or in Antarc-tic ice typically have thresholds of the order of tens of GeV, which makes them sensitivemainly to heavy neutralinos (above 100 GeV) [79]. In DarkSUSY, the low energy cut-off canbe set.
Final states which give a hard neutrino spectrum (such as heavy quarks, τ leptons andW or Z bosons) are usually more important than the soft spectrum arising from light quarksand gluons.
Neutralinos are steadily being trapped in the Sun or Earth by scattering, whereas anni-hilations take them away. The balance between capture and annihilation has the solution forthe annihilation rate implemented in DarkSUSY
ΓA =Cc
2tanh2
(
t
τ
)
, (40)
where the equilibration time scale τ = 1/√CcCa, with Cc being the capture rate and Ca
being related to the annihilation efficiency. In most cases for the Sun, τ is much smallerthan a few billion years, and therefore equilibrium is often a good approximation (N(t) = 0).This means that it is the capture rate which is the important quantity that determines theneutrino flux. For the Earth, τ is, on the other hand, usually of the same order as, or muchlarger than, the age of the solar system, and equilibrium has often not occurred. In eithercase, in the program we keep the exact formula (40) (except for the modifications needed fora Damour-Krauss population of WIMPs, discussed below).
For the actual capture rate calculations we have several expressions implemented in Dark-
SUSY. As a default, we use the full expressions given in appendix A of [80] where we numer-ically integrate over the velocity distribution. To speed-up the calculations, it is possible toperform this integration only once and use a saved tabulated version for subsequent calls.In the capture rate calculations we also need the density profile of the Earth and the Sunand the chemical composition as a function of radius. For the Sun we use the solar modelBP2000 [81], complemented with the estimates of the mass fractions of the heavier elementsfrom [82]. For the Earth, we use the density profile of [83] with the compositions given in[84] (see [54] for a table of these). For comparison, the approximate capture rate expressionsin [6] are also available.
16
The capture rate induced by scalar (spin-independent) interactions between the neutrali-nos and the nuclei in the interior of the Earth or Sun is the most difficult one to compute, sinceit depends sensitively on the Higgs mass, form factors, and other poorly known quantities.However, this spin-independent capture rate calculation is the same as for direct detectiontreated in Section 6.1. Therefore, there is a strong correlation between the neutrino flux ex-pected from the Earth (which is mainly composed of spin-less nuclei) and the signal predictedin direct detection experiments [79, 85].
Due to the low counting rates for the spin-dependent interactions in terrestrial detectors,high-energy neutrinos from the Sun constitute a competitive and complementary neutralinodark matter search. Of course, even if a neutralino is found through direct detection, it willbe extremely important to confirm its identity and investigate its properties through indirectdetection. In particular, the mass can be determined with reasonable accuracy by looking atthe angular distribution of the detected muons [86, 87].
The capture rate in the Earth is dominated by scalar interactions, where there maybe kinematic and other enhancements, in particular if the mass of the neutralino almostmatches one of the heavy elements in the Earth. As shown by Gould [88], the Earth doesnot dominantly capture WIMPs from the halo directly. Instead, the Earth captures WIMPsthat, due to gravitational interactions, have diffused around and become bound to the solarsystem. However, solar depletion of these bound WIMPs could be an important effect [89],and as a default, DarkSUSY uses a new estimate of the velocity distribution in the solarsystem, where these solar depletion effects have been included [54]. It is possible to changeto a standard Gaussian distribution if the user prefers.
There is also a possibility that there exists a special population of WIMPs, the Damour-Krauss population [90], arising from WIMPs that have just skimmed the Sun’s surface. Asan optional choice, this population can be included in the calculation [66]. The enhancementcaused by the new population is only important for a neutralino mass less than 150 - 170GeV (the exact number depending on details about the angular momentum distribution [66]).The total capture rate is computed according to the formulas in [66], which take into accountthat the annihilation rates from the Earth will in general depend on time in a different waythan the simple result in Eq. (40).
6.4 Indirect searches through antimatter signals
Pair annihilation of neutralinos in the Galactic halo produces the same amounts of matter andantimatter. As antimatter seems to be scarce in the Universe, apparently with no standardprimary sources, there is a chance that by measuring antimatter in cosmic ray fluxes onemay find evidence of the existence of dark matter particles (see, e.g., [91, 92, 93, 94, 95,96, 97, 98, 99]). In the current release of the code we consider neutralino induced fluxesof antiprotons, positrons and antideuterons. To produce estimates of such fluxes, there areseveral steps one needs to follow: i) Evaluate for each dark matter candidate the probabilityfor a pair annihilation to take place locally in space, i.e. compute 1/2 (ρχ(~x )/mχ)
2σannv;8
ii) Estimate the production rate of a given species by folding together, for each model, thebranching ratios for the annihilation into a given two-body final state with the Monte Carlosimulation of the hadronization and/or decay of that state, as described in Section 6.2.1(except for D sources where we have implemented the prescription suggested in Ref. [98] toconvert from the p yield); iii) Propagate these sources through the Galactic magnetic fieldsto make predictions for the induced cosmic ray fluxes at the Sun’s location in the Galaxy; iv)Include the effect of solar modulation to propagate the fluxes from the interstellar mediumto our location inside the solar system. The implementation in DarkSUSY is written in a
8The origin of the factor of 1/2, missing in earlier versions of DarkSUSY, is explained in Sect. 4.4.
17
modular format which allows the user to eventually modify and/or replace each of the foursteps above independently.
At tree level, the relevant final states of neutralino pair annihilations for p and D produc-tion are qq, W+W−, Z0Z0, W+H−, ZH0
1 , ZH02 , H
01H
03 and H0
2H03 ; among the qq states we
have included in DarkSUSY all the heavier quarks (c, b and t). In addition, we have includedthe Zγ ([100]) and the 2 gluon ([101]; [102]) final states which occur at one loop-level. Thesame list of final states is implemented for positrons, with the addition of ℓ+ℓ−. The e+e−
final state gives rise to a positron monochromatic source (however this is negligible in mostparticle physics setups, and tends to be smeared out by propagation).
To model the propagation we have considered a semi-empirical diffusion equation [103,104], in the steady state approximation, applied to a two-dimensional model with cylindricalsymmetry and with free escape boundary conditions. The parameters in such a model shouldbe fixed in agreement with values inferred from available cosmic ray data in analogous propa-gation models (see, e.g., the Galprop package [105]). For simplicity, rather than consideringthe most general setup, we restrain to cases in which, still including all relevant effects, theGreen function of the transport equation can be derived analytically, so that we can avoidthe CPU time-consuming problem of having to solve a partial differential equation for eachdark matter candidate. Therefore, we do not include reacceleration effects but mimic themthrough a diffusion coefficient which has a broken power law in rigidity as functional form.Also , for antiprotons [96] and antideuterons we neglect energy losses (whenever a scatteringwith a nucleus takes place the particle is removed), while for positrons [97] we consider anaverage over space of the energy loss effect due to inverse Compton on starlight and thecosmic microwave background, and in addition the synchrotron radiation from the effects ofthe galactic magnetic field. For comparison, we allow also the option to treat the propagationof antiprotons according to the propagation models by Chardonnet et al. [94] and Bottino etal. [95], while for the positron we have implemented the option to use the leaky-box treat-ment given by Kamionkowski and Turner [93] or the numerical Green functions derived byMoskalenko and Strong [106] with the Galprop code [105] (the latter two cases howevercannot be interfaced to a generic axisymmetric dark matter density profile, as for our defaultpropagation model).
Given a set of parameters for the propagation model, and a given neutralino numberdensity profile or a given probability distribution of small clumps, the effect of propagation, inthe cases we have considered, can be factorized out into effective energy-dependent “diffusiontimes”, τp(T ), τD(T ) and τe+(T ), which are independent of the parameters defining theparticle physics setup for the dark matter candidate. In some cases, such as when consideringvery large samples of neutralino candidates or when implementing singular halo profilesfor which the computation of the diffusion times can be very CPU time-consuming, it isadvantageous to exploit the option provided by DarkSUSY to tabulate the diffusion timesover a given energy range (optionally saving the tabulation to disk for later use) and use aninterpolation between tabulated values, rather than linking directly to the computation foreach dark matter candidate.
Finally, regarding solar modulation, we implement the one parameter model based on theanalytical force-field approximation by Gleeson & Axford [107] for a spherically symmetricmodel. This approach is expected to be slightly less accurate than, e.g., the full numericalsolution of the propagation equation of the spherically symmetric model ([108]), but again itis much less CPU time-consuming. DarkSUSY allows an output with both solar modulatedand local interstellar fluxes, and the latter can eventually be solar modulated by the userwith more sophisticated methods. For the positrons we allow for the option to reduce theeffects of solar modulation by considering the positron fraction, e+/(e++e−), rather than theabsolute positron fluxes. In this case an estimate of the background is needed: in DarkSUSY
18
we provide background e+ and e− fluxes taken from [106].
6.5 Gamma rays
Gamma-rays have a low enough cross section on gas and dust that the Galaxy is essentiallytransparent to them (except perhaps in the innermost part, very close to the region wherea massive black hole is inferred); absorption by starlight and infrared background becomesefficient only for very far away sources and high-energy photons.
The bulk of the gamma-rays from neutralino annihilations arise in the decay of neutral pi-ons produced in the fragmentation processes initiated by tree level final states [109, 60, 110](analogously to the other halo signals, in DarkSUSY we include all tree level final statesand make use of a Monte Carlo simulation for fragmentation and decay processes, see Sec-tion 6.2.1). However, π0 production is common also to other astrophysical processes, andthis may turn out to be a limiting factor to disentangle dark matter sources. At the sametime, however, a relevant gamma-ray contribution may arise directly (at one-loop level) intwo body final states; although such photons are much fewer than those from π0 decays theyhave a much better signature: neutralinos annihilating in the galactic halos move with avelocity of the order v/c ∼ 10−3, hence these outgoing photons (as any particle in any of theallowed two body final states) will then be nearly monochromatic, with energy of the orderof the neutralino mass [111, 112, 113, 102, 100, 60]. There is no other known astrophysi-cal source with such a signature: the detection of a line signal out of a spectrally smoothgamma-ray background would be a spectacular confirmation of the existence of dark matterin form of exotic massive particles.
6.5.1 Sources and fluxes
Following the discussion in [60], the monochromatic gamma-ray flux measured in a detectorwith angular acceptance ∆Ω is:
Φγ(ψ,∆Ω) = 0.94 · 10−11
(
Nγ vσX0γ
10−29 cm3s−1
)(
10GeV
Mχ
)2
〈J (ψ)〉∆Ω ×∆Ω cm−2 s−1 sr−1 ,
(41)where ψ is an angle to label the direction of observation and where Nγ = 2 for χχ → γ γ,Nγ = 1 for χχ→ Z γ. Here the dimensionless line-of-sight dependent function is
J (ψ) =1
8.5 kpc·(
1
0.3GeV/cm3
)2 ∫
line of sight
ρ2χ(l) d l(ψ) , (42)
and its angular average over the resolution solid angle ∆Ω is
〈J (ψ)〉∆Ω =1
∆Ω
∫
∆Ω
dΩ′J (ψ′) , (43)
Analogously, the gamma-ray flux with continuum energy spectrum is obtained by replac-ing Nγ vσX0γ with
∑
f dNfγ /dE vσf , where the sum is over all tree level final states.
Finally, the formalism we introduced can be used also to estimate the flux in the simplecase of a single source which, for a given detector, can be approximated as point-like. If sucha source is in the direction ψ at a distance d, Eq. (43) becomes:
〈J (ψ)〉∆Ω =1
8.5 kpc·(
1
0.3GeV/cm3
)2
· 1
d2· 1
∆Ω
∫
d3r ρ2χ(~r) (44)
where the integral is over the extension of the source (much smaller than d).
19
Several targets have been discussed as sources of gamma-rays from the annihilation ofdark matter particles. An obvious source is the dark halo of our own galaxy [114] and inparticular the Galactic center, as the dark matter density profile is expected, in most models,to be peaked towards it, possibly with huge enhancements close to the central black hole. TheGalactic center is an ideal target for both ground- and space-based gamma-ray telescopes.As satellite experiments have much wider field of views and will provide a full sky coverage,they will test the hypothesis of gamma-rays emitted in clumps of dark matter which maybe present in the halo [115, 110, 116, 117]. Another possibility which has been considered isthe case of gamma-ray fluxes from external nearby galaxies [118]. Furthermore, it has beenproposed to search for an extragalactic flux originated by all cosmological annihilations ofdark matter particles [119, 120].
DarkSUSY is suitable to compute the gamma-ray flux from all these (and possibly other)sources. Two cases are fully included in the package: assuming that neutralinos are smoothlydistributed in the Galactic halo with ρχ equal to the dark matter density profile, in DarkSUSY
Eq. (43) is computed for a specified halo profile and any given ψ and ∆Ω [60]. The secondoption deals with the case of a portion of dark matter being in the form of clumps, each ofwhich is treated as a non-resolvable source in the detector, distributed in the Galaxy accordingto a probability distribution which can be specified by the user; in DarkSUSY the defaultprobability distribution stems from the assumption that it follows the dark matter densityprofile (see [116] for details). It is straightforward to extend this to all other astrophysicalsources; in case of cosmological sources one has just to pay attention to include redshifteffects and absorption on starlight and infrared background, see [120].
The case of the possible enhancement, a “spike” in the vicinity of the galactic center [121]should be kept in mind. However, since there is no consensus in the literature [122, 123, 124]about important quantities for the calculation such as the magnetic field radial profile andthe optical depth for synchrotron self-absorption, we have chosen not to include routines forthe effects of this possible enhancement of gamma rays and neutrinos. And the very existenceof a spike is dependent on fine details, still unknown for the Milky Way [125, 126].
6.5.2 Gamma rays with continuum energy spectrum
The gamma-ray flux produced in neutralino annihilations through π0 decays can be large butin general lacks distinctive features. The photon spectrum in the process of a pion decayinginto 2γ is, independent of the pion energy, peaked at half of the π0 mass, about 70 MeV,and symmetric with respect to this peak if plotted in logarithmic variables (i.e. dNγ/dE vs.logE). Of course, this is true both for pions produced in neutralino annihilations and, e.g.,for those generated by cosmic ray protons interacting with the interstellar medium.
When considered together with the cosmic ray induced Galactic gamma-ray background,the neutralino induced signal looks like a component analogous to the secondary flux due tonucleon nucleon interactions: it is dwarfed by the bremsstrahlung component at low energy,while it may be the dominant contribution at energies above 1 GeV or so. In DarkSUSY
the continuum gamma flux from all annihilation channels is computed and may be easilyobtained for a given energy or energy threshold.
6.5.3 χχ→ γγ
At the one-loop level, it is possible to get two-body final states containing one or two photons,with a distinctive signature which may provide a “smoking gun” for dark matter. Thedrawback is of course that the processes are loop-suppressed, so one probably needs a halowith a large central concentration, or small-scale structure (“clumps” of dark matter) todetect a signal.
20
In DarkSUSY the full expression for the annihilation cross section of the process
χ01 + χ0
1 → γ + γ (45)
is computed at full one loop level, in the limit of vanishing relative velocity of the neutralinopair, i.e. the case of interest for neutralinos in galactic halos; the outgoing photons have anenergy equal to the mass of χ0
1:Eγ = mχ. (46)
The total amplitude is implemented in DarkSUSY as the sum of the contributions obtainedfrom four different classes of diagrams:
A = Aff + AH+ + AW + AG,
where the indices label the particles in the internal loops, i.e., respectively, fermions andsfermions, charged Higgs and charginos, W-bosons and charginos, and, in the gauge we chose,charginos and Goldstone bosons. For every A term, real and imaginary parts are separatelycomputed; the full set of analytic formulas are given in [102], following the notation of [113],where some of the contributions were first computed. They are rather lengthy expressionswith non trivial dependences on various combinations of parameters in the MSSM.
The branching ratio for neutralino annihilations into 2γ is typically not larger than 1%,with the largest values of vσ2γ , for neutralinos with a cosmologically significant relic abun-dance, in the range 10−29–10−28 cm3s−1. Such values may be large enough for the discoveryof this signal in upcoming measurements; at the same time it should be kept in mind thatvery low values for the cross section are possible as well.
In the very high mass range above a TeV, it has been suggested that the line rates maybe very much larger due to binding effects and resonant conversion between neutralinos andcharginos [127]. In the present version of DarkSUSY we have not included these effects,however.
6.5.4 χχ→ Zγ
The process of neutralino annihilation into a photon and a Z0 boson [100]
χ01 + χ0
1 → γ + Z0 (47)
also gives a nearly monochromatic line (with a small smearing caused by the finite widthof the Z0 boson, in any case unlikely to be important for current or proposed gamma rayexperiments), with energy
Eγ =Mχ − m2Z
4Mχ. (48)
The steps followed in DarkSUSY to compute the cross section are essentially the sameas those described for the 2γ case. Again the total amplitude is obtained by summing thecontribution from four classes of diagrams and by splitting for each of them real and imaginaryparts. The analytic formulas were derived in [100], and are much less compact than thoseobtained for the process of neutralino annihilation into two photons.
The maximum value of vσZγ , for neutralinos with a cosmologically significant relic abun-dance, is around 2 · 10−28 cm3s−1 and occurs for nearly pure higgsinos. In the heavy massrange, the value of vσZγ reaches a plateau of around 0.6 · 10−28 cm3s−1. This interestingeffect of a non-diminishing cross section with Higgsino mass (which is due to a contributionto the real part of the amplitude) is also valid for the 2γ final state in the correspondinglimit, with a value of 1 · 10−28 cm3s−1 [102]. Since the gamma-ray background drops rapidly
21
with increasing photon energy, these processes may be interesting for detecting dark matterneutralinos near the upper range of allowed neutralino masses.
Whenever the lightest neutralino contains a significant Wino or Higgsino component thevalue of vσZγ may be as large as, or even larger than, twice the value of vσ2γ . It is thereforeusually not a good approximation to neglect the Zγ state compared to 2γ.
6.6 Neutrinos from the halo
Usually, the flux of neutrinos from annihilation of neutralinos in the Milky Way halo istoo small to be detectable, but for some clumpy or cuspy models, or for annihilation in apossible spike around the central black hole, it might be detectable. The calculation of theneutrino-flux follows closely the calculation of the continuous gamma ray flux, with the mainaddition that neutrino interactions close to the detector are also included. Hence, both theneutrino flux and the neutrino-induced muon flux can be obtained. The neutrino to muonconversion rate in the Earth can also be obtained. The neutrino rate from other sources thanthe interior of the Earth or the Sun is generally negligible. If there would exist a spike at thegalactic center [121], there may in some cases be a detectable flux. These neutrino rates areconstrained by existing limits on the gamma-ray flux [121, 128].
7 Examples of results obtained with DarkSUSY
We will here go through a set of benchmark models as examples of the performance ofDarkSUSY. We will consider two popular setups. We will start with a set of mSUGRA modelsand then turn to more general MSSM models. We will here use the default DarkSUSY setup,in particularan isothermal sphere with a Maxwell-Boltzmann velocity distribution for thehalo model.
7.1 Benchmark models in mSUGRA
We will consider here some of the benchmark models from Battaglia et al. [129]. In table 1we list the properties of the selected models, as derived by DarkSUSY using to the ISASUGRA7.69 package to describe the renormalization group running of the theory from the grandunification scale to the low energy scale. The models we are focusing on are those with a topmass of 175 GeV and that are still viable with ISASUGRA 7.69. As the table in [129] wasproduced with ISASUGRA 7.67, some differences occur due to different versions of the codes,e.g. model M is no longer physical in ISASUGRA 7.69, and thus not included here.
Our results for the relic density and for the SUSY contributions to B(b→ sγ) and to theanomalous magnetic moment of the muon aµ, agree reasonably well with those in [129]. Theexpected sensitivities of future neutrino telescopes are of the order of 20 events km−2 yr−1 forthe Earth and 50–1000 events km−2 yr−1 for the Sun; we see in the table that all benchmarkmodels, unfortunately, produce much lower fluxes. The outreach of future direct detectionexperiments is expected to be of the order of 10−9 pb for the spin-independent scatteringcross section; we see that some of models we are considering are potentially detectable. Thecross section for annihilation into gamma rays (times the number of photons produced) arealso given for these models; the detectability depends strongly on the halo profile, but one canin general say that these cross sections are too low to be seen in current data, unless the haloprofile is very cuspy towards the galactic center. The e+ fluxes are here given in the energy bin6.0–8.9 GeV of the HEAT 94+95 [130] experiment. The measured e+ flux is (7.2±1.2)×10−6
GeV−1 cm−2 s−1 sr−1, i.e. the predicted fluxes are much lower than the measured one. Forthe antiprotons, we choose, as an example, to show the predicted (average) flux in the energy
22
Model A’ B’ C’ D’ G’ H’ I’ J’ K’ L’
m1/2 [GeV] 600 250 400 525 375 935 350 750 1300 450
m0 [GeV] 107 57 80 101 113 244 181 299 1001 303tan β 5 10 10 10 20 20 35 35 46 47
sign(µ) + + + - + + + + - +
mχ [GeV] 242.8 94.9 158.1 212.4 148.0 388.4 138.1 309.1 554.2 181.0
B(b → sγ) × 104 3.96 3.88 3.95 4.39 3.67 3.84 3.33 3.84 4.40 3.62
aµ × 1010 1.3 18 6.7 −4.4 16 2.7 33 7.4 −3.3 26
Incl. coanns e2 µ2 τ1 χ01
e2 µ2 τ1 χ01
e2 µ2 τ1 χ01
e2 µ2 τ1 χ01
e2 µ2 τ1 χ01
e2 µ2 τ1 χ01
τ1 χ01
e2 µ2 τ1 χ01
χ01
τ1 χ01
Ωχh2 0.0929 0.1213 0.1149 0.0864 0.1294 0.1629 0.1319 0.0966 0.0863 0.0988
Φ⊕µ best [km−2 yr−1] 2.46 · 10−10 1.50 · 10−5 9.76 · 10−9 3.76 · 10−14 3.36 · 10−7 1.51 · 10−12 1.89 · 10−5 2.59 · 10−10 4.59 · 10−16 1.07 · 10−5
Φ⊕µ gauss [km−2 yr−1] 3.03 · 10−9 5.18 · 10−5 7.15 · 10−8 3.92 · 10−13 2.27 · 10−6 3.50 · 10−11 1.17 · 10−4 4.40 · 10−9 1.48 · 10−14 9.20 · 10−5
Φ⊙µ [km−2 yr−1] 1.22 · 10−2 1.47 · 101 5.50 · 10−1 3.01 · 10−2 3.33 · 100 1.76 · 10−4 4.45 · 100 1.44 · 10−2 3.42 · 10−3 2.32 · 100
σSIp std [pb] 3.02 · 10−10 5.69 · 10−9 8.22 · 10−10 2.39 · 10−12 1.98 · 10−9 8.38 · 10−11 7.93 · 10−9 2.81 · 10−10 6.80 · 10−14 5.94 · 10−9
σSDp std [pb] 2.13 · 10−7 1.06 · 10−5 1.55 · 10−6 5.02 · 10−7 2.33 · 10−6 8.64 · 10−8 3.31 · 10−6 2.06 · 10−7 3.64 · 10−8 1.37 · 10−6
σSIp dn [pb] 3.02 · 10−10 5.71 · 10−9 8.23 · 10−10 2.63 · 10−12 1.99 · 10−9 8.40 · 10−11 7.99 · 10−9 2.83 · 10−10 7.95 · 10−14 5.97 · 10−9
σSDp dn [pb] 2.09 · 10−7 1.05 · 10−5 1.53 · 10−6 4.93 · 10−7 2.30 · 10−6 8.51 · 10−8 3.27 · 10−6 2.04 · 10−7 3.60 · 10−8 1.36 · 10−6
Nγ cont.(σv) [10−29 cm3 s−1] 120 782 195 63.6 1032 86.5 6303 930 70803 18739
2(σv)γγ [10−29 cm3 s−1] 1.0 2.6 1.7 1.3 1.4 0.35 0.77 0.33 0.017 0.39
(σv)Zγ [10−29 cm3 s−1] 0.17 0.31 0.26 0.19 0.18 0.051 0.083 0.040 0.0022 0.037
Φe+
[GeV−1 cm−2 s−1 sr−1] 1.5 · 10−11 5.4 · 10−10 7.0 · 10−11 2.2 · 10−11 4.1 · 10−10 7.7 · 10−12 2.3 · 10−9 1.1 · 10−10 3.7 · 10−9 4.7 · 10−9
Φp [GeV−1 cm−2 s−1 sr−1] 6.0 · 10−11 2.2 · 10−9 1.5 · 10−10 2.1 · 10−11 9.0 · 10−10 7.2 · 10−12 6.7 · 10−9 1.4 · 10−10 2.5 · 10−9 1.0 · 10−8
ΦD [GeV−1 cm−2 s−1 sr−1] 2.3 · 10−15 9.1 · 10−14 5.8 · 10−15 8.7 · 10−16 3.6 · 10−14 2.7 · 10−16 2.6 · 10−13 5.1 · 10−15 9.0 · 10−14 4.0 · 10−13
Table 1: Relic density and various rates for the benchmark models of [129]. There are five free parameters in mSUGRA: tanβ, sign(µ),m1/2, m0 and A0. The latter three are the unification values (at the grand unification scale) of the soft supersymmetry breaking fermionicmass parameters, scalar mass parameters and trilinear scalar coupling parameters, respectively. All the benchmark models have A0 = 0.We have here used ISASUGRA 7.69 for the RGE-running (but have not taken the b, t, and τ Yukawa couplings from ISASUGRA). Theneutrino-induced muon fluxes from the Earth and the Sun are for a threshold of 1 GeV. ‘best’ refers to the suppressed fluxes resulting fromthe estimate in [54] and ‘gauss’ refers to the usual approximation of a Gaussian velocity distribution of neutralinos for capture in the Earth.The scattering cross sections are given here evaluated both with the standard expressions (labelled ‘std’) and with the Drees and Nojiriexpressions [70] (labelled ‘dn’). The continuum γ’s are given in terms of the number of photons times the cross section, Nγ cont.(σv), and
refers to γ’s above 1 GeV. The e+ flux is the average solar modulated flux in the energy range 6.0–8.9 GeV, i.e. in one of the HEAT 94-95’sbins[130]. The p flux is the average solar modulated flux in the energy range 0.56–0.78 GeV, i.e. in one of the BESS 98 bins [131]. TheD flux is the average flux in the energy range 0.1–0.4 GeV, as applicable to e.g. the proposed GAPS probe [132]. In this case, the flux isthe average of the solar modulated fluxes at solar minimum and maximum. Parameters other than those mentioned above are set to theirdefault values as described in the text.
23
range 0.56–0.78 GeV, which is one of the BESS 98 [131] bins. The measured flux in thisbin is (1.23+0.38
−0.33)× 10−6 GeV−1 cm−2 s−1 sr−1, i.e. the measured flux is much higher thanthe predicted one in this energy range. For the antideuterons, we show the expected fluxin the energy range 0.1–0.4 GeV, which is reasonable for the proposed GAPS probe [132].For the antideuterons, there is essentially no background and the sensitivity is thus given bythe ability to detect one antideuteron. For GAPS this corresponds to a sensitivity of about2.6× 10−13 GeV−1 cm−2 s−1 sr−1. Hence, two of these benchmark models have high enoughfluxes for being just about detectable in this way. For further examples of rates in mSUGRA,as calculated with DarkSUSY , see e.g. [133].
7.2 Benchmark models in MSSM
We will now turn to models in the MSSM framework as generated by fixing free parametersat the weak scale. We refer to the setup with seven free parameters, i.e. µ, M2, mA, tanβ,m0, At and Ab, we described in Section 2. We will here show an example of results thatcan be obtained with a simple scan over this parameter space. We have generated 5000models assuming: µ ∈ [−3000, 3000] GeV, M2 ∈ [−3000, 3000] GeV, mA ∈ [100, 1000] GeV,tanβ ∈ [1, 55], m0 ∈ [100, 5000] GeV, At ∈ [−3, 3]m0 and Ab ∈ [−3, 3]m0. We have thenselected a few sample models with a relic density in the range 0.09 ≤ Ωχh
2 ≤ 0.11 and withreasonably high detection rates; these are the first ten models in Table 2. In addition to thisscan, we have also made a small scan of 300 models in which we required the mass of theCP -odd Higgs boson, A, to be in the range mA ∈ [90, 150] GeV; model number 11 in Table2 has been chosen from this latter scan to illustrate that higher-rate models are possible tofind with these kind of dedicated scans.
As seen in the table, the rates are typically slightly higher than for the mSUGRA bench-mark models considered above. Please remember though, that this set of MSSM benchmarkmodels is achieved with a rather small scan over the MSSM parameter space. Models witheven higher rates would be possible to find with more extensive scans of the parameter space.
Many of our models produce fluxes from the Sun that exceed the future sensitivity of50–1000 events km−2 yr−1 and hence would be detectable. For the Earth, on the other hand,the fluxes in this set of models are typically much lower than the projected future limit of 20events km−2 yr−1. Most of the selected models are potentially detectable with future directdetection experiments as they have a spin-independent scattering cross section larger than10−9 pb. Note the complementarity between the direct detection signal and the neutrinoflux from the Sun. For example, for model 9, the spin-independent scattering cross sectionis very close to the sensitivity of future detectors, whereas the neutrino-flux from the Sunis clearly above expected future sensitivities of neutrino telescopes like Antares or IceCube.The gamma-ray yields are in general larger than for the mSUGRA benchmark points, but theissue regarding detectability is still difficult to address as it is strongly affected by the halomodel dependence. The positron and antiproton fluxes are generally enhanced as well, butstill well below measured fluxes. Finally, there are several models among those we selectedwhich have an antideuteron flux in the energy range 0.1–0.4 GeV exceeding 2.6 × 10−13
GeV−1 cm−2 s−1 sr−1, the expected sensitivity of the GAPS detector [132].
7.3 Discussion
The potential of DarkSUSY package has been illustrated here, for a few sample models andin two popular scenarios. The code provides the state of the art calculation of the neutralinorelic abundance, and allows for the estimate of several quantities of interest for dark mattersearches. It has been shown that the code has a very flexible structure in the definition ofthe supersymmetric dark matter candidate, as well as a rather broad freedom in the choice
24
Model 1 2 3 4 5 6 7 8 9 10
µ [GeV] 441.9 −203.0 218.9 153.8 996.6 −109.8 122.2 −1020.2 512.2 −336.1M2 [GeV] −785.4 329.0 −361.0 −213.3 2446.9 201.6 282.0 2895.0 776.2 −438.4mA [GeV] 925.9 507.8 950.7 703.4 436.5 856.9 587.9 753.2 636.0 104.4
tan β 9.7 8.3 15.7 38.4 13.1 13.4 3.3 14.7 3.0 27.7m0 [GeV] 2675.2 4793.8 4078.7 4249.6 1573.0 4511.5 1404.5 4001.4 1084.7 589.8At/m0 1.14 −0.75 2.39 −1.10 −2.43 0.63 2.32 −0.58 2.46 2.50Ab/m0 −1.92 2.90 −1.02 −1.62 −0.77 0.47 −1.81 0.62 1.79 0.98
mχ [GeV] 382.5 151.7 165.2 92.2 98.8 73.2 78.0 1017.8 378.4 212.9Zg 0.788 0.690 0.675 0.672 0.020 0.377 0.298 0.005 0.907 0.917
B(b → sγ) × 104 4.01 4.45 3.97 4.37 4.22 4.12 4.42 4.20 4.51 2.82
aµ × 1010 −0.63 −0.27 −0.59 −1.26 0.61 −0.45 0.28 −0.24 0.50 15.87
Incl. coanns χ+2
χ01,2,3
χ+2
χ01,2,3
χ+2
χ01,2,3
χ+2
χ01
t1 χ+2
χ01,2,3
χ+2
χ01
χ+2
χ01
χ+2
χ01,2,3
χ+2
χ01,2,3
χ+2
χ01,2
Ωχh2 0.0926 0.0982 0.0960 0.0988 0.0918 0.1000 0.0963 0.1032 0.0977 0.0968
Φ⊕µ best [km−2 yr−1] 7.62 · 10−6 1.94 · 10−5 1.05 · 10−4 5.63 · 10−5 7.60 · 10−6 3.30 · 10−5 3.66 · 10−3 1.62 · 10−8 4.67 · 10−5 2.17 · 101
Φ⊕µ gauss [km−2 yr−1] 1.62 · 10−4 1.27 · 10−4 7.56 · 10−4 1.72 · 10−4 2.45 · 10−4 5.58 · 10−5 7.43 · 10−3 5.26 · 10−7 9.79 · 10−4 2.04 · 102
Φ⊙µ [km−2 yr−1] 6.06 · 101 1.56 · 103 1.29 · 103 3.23 · 103 3.69 · 100 1.27 · 103 6.77 · 102 3.08 · 10−1 1.74 · 101 3.66 · 103
σSIp std [pb] 4.08 · 10−9 1.69 · 10−9 4.29 · 10−9 1.41 · 10−9 6.64 · 10−9 6.23 · 10−9 5.95 · 10−8 3.22 · 10−10 1.01 · 10−8 4.14 · 10−6
σSDp std [pb] 4.90 · 10−5 4.31 · 10−4 3.59 · 10−4 9.16 · 10−4 1.80 · 10−6 2.63 · 10−3 8.83 · 10−4 5.26 · 10−7 6.10 · 10−6 3.47 · 10−5
σSIp dn [pb] 4.08 · 10−9 1.69 · 10−9 4.29 · 10−9 1.41 · 10−9 6.75 · 10−9 6.23 · 10−9 5.92 · 10−8 3.22 · 10−10 1.01 · 10−8 4.14 · 10−6
σSDp dn [pb] 4.90 · 10−5 4.31 · 10−4 3.59 · 10−4 9.16 · 10−4 1.79 · 10−6 2.63 · 10−3 8.83 · 10−4 5.26 · 10−7 6.02 · 10−6 3.40 · 10−5
Nγ cont.(σv) [10−29 cm3 s−1] 7.23 · 104 2.90 · 104 3.05 · 104 1.39 · 104 4.14 · 104 3.93 · 102 6.20 · 102 3.45 · 104 8.04 · 104 4.64 · 104
2(σv)γγ [10−29 cm3 s−1] 0.073 0.57 0.64 0.67 3.21 1.21 3.39 5.07 0.0077 0.059
(σv)Zγ [10−29 cm3 s−1] 0.44 1.95 2.21 1.49 4.53 1.22 3.36 4.91 5.41 0.11
Φe+
[GeV−1 cm−2 s−1 sr−1] 5.65 · 10−9 9.86 · 10−7 8.70 · 10−7 1.32 · 10−6 2.48 · 10−8 5.97 · 10−8 8.33 · 10−8 1.91 · 10−8 4.20 · 10−7 7.78 · 10−7
Φp [GeV−1 cm−2 s−1 sr−1] 1.25 · 10−8 3.95 · 10−8 3.17 · 10−8 8.16 · 10−8 1.40 · 10−10 3.13 · 10−9 3.68 · 10−9 8.09 · 10−11 1.25 · 10−8 2.05 · 10−8
ΦD [GeV−1 cm−2 s−1 sr−1] 4.52 · 10−13 8.15 · 10−13 5.34 · 10−13 3.85 · 10−12 3.07 · 10−15 1.60 · 10−13 1.57 · 10−13 2.06 · 10−15 6.55 · 10−13 4.33 · 10−13
Table 2: Same as Table 1, but for the MSSM benchmark models of our sample scan.
25
of the relevant astrophysical setups. Outputs are in simple and general formats, which, aswe have shown, can be very easily compared to current and future sensitivities. Some trendson predictions can be extracted from benchmark models, as we have done for some of themSUGRA models proposed in [129], and with sample models selected according to their relicabundance in a more general low energy scan (with the latter being more promising than theformer). One should keep in mind however that firmer statements are possible just in lightof dedicated and more extensive scans.
8 Conclusions
We have here described the computer packageDarkSUSY, that can be used to calculate variousquantities of interest for supersymmetric dark matter searches. We have gone through greatefforts and used state-of-the-art techniques to obtain a package that can deliver very accurateresults in a flexible setting. We also believe that this package can be of great use for thephysics community.
In this paper we have described briefly the ingredients of DarkSUSY and shown, with someexamples, what one can calculate with it. We encourage the reader to download DarkSUSY
and start using it.
Acknowledgements
We thank all our users of previous releases of DarkSUSY for their feedback. The work ofL.B. was partially supported by the Swedish Research Council (VR). J.E. was supported bythe Swedish Research Council. P.U. was supported in part by the RTN project under grantHPRN-CT-2000-00152 and by the Italian INFN under the project “Fisica Astroparticellare”.
A Included coannihilations
In this Appendix, we tabulate the coannihilation processes computed in DarkSUSY. For moredetails, see [46].
In the table, we list all 2 → 2 tree-level coannihilation processes with sfermions, charginosand neutralinos. All the processes are included in the DarkSUSY code.
It should be noted that we have not included all flavour-changing charged current dia-grams. The DarkSUSY vertex code for the charged current couplings is written in a gen-eral form that includes all possible flavour-changing (and flavour-conserving) vertices. Theflavour-conserving couplings are much larger than the flavour-changing. For the sfermioncoannihilations with charged currents we only take the flavour-conserving contributions, whilefor the chargino coannihilations we include the flavour-changing contributions as well. In afuture version of DarkSUSY, we may as well include the flavour-changing processes for thesfermion coannihilations, even if they are not expected to be important.
We have used the notation f for sfermions and f for fermions. Whenever the isospin ofthe sfermion/fermion is important, it is indicated by an index u (T3 = 1/2) or d (T3 = −1/2).The sfermions have an additional mass eigenstate index, that can take the values 1 and 2(except for the sneutrinos which only have one mass eigenstate). A further complicationto the notation is when the sfermions and fermions in initial, final and exchange state canbelong to different families. Primes will be used to indicate when we have this freedom tochoose the flavour. So, e.g. fu and fu will belong to the same family while fu and f ′
u canbelong to the same or to different families. Note that the colour index of (s)quarks as wellas gluons (g) and gluinos (g) is suppressed.
26
Besides the sfermions we also have neutralinos and charginos in the initial states. Thenotation used for these are the following. The neutralinos are denoted by χ0
j with the index
running from 1 to 4. The charginos are similarly denoted χ±j with the index taking the values
1 and 2.In the table, a common notation is introduced for gauge and Higgs bosons in the final
state. We denote these with B with an upper index indicating the electric charge. So B0
means H01 , H
02 , H
03 , Z, γ and g while B± is H± and W±. We will use additional lower indices
m and n when we have more than one boson in the final state. Thus indicating that thebosons can be either different or identical. Note that the case of two different bosons alsoincludes final states with one gauge boson and one Higgs boson.
The table has been made very general. This means that when a set of initial and finalstate (s)particles has been specified, the given process might not run through all the exchangechannels listed for the generic process. Exceptions occur whenever an exchange (s)particledoes not couple to the specific choice of initial and/or final state. As an example we see thatsince the photon does not couple to neutral (s)particles, none of the exchange channels listedfor the generic process fi+χ
0j → B0+f actually exist for the specific process ν+χ0 → γ+ν.
All these exceptions can be found in the extended tables in Ref. [134]. Also note that thelist of processes is not complete with respect to trivial charge conjugation. For each processof non-vanishing total electric charge in the initial state there exists another process whichis obtained by charge conjugation.
27
DiagramsProcess s t u p
χ0iχ
0j → B0
mB0n H0
1,2,3, Z χ0k χ0
l
χ0iχ
0j → B−
mB+n H0
1,2,3, Z χ+k χ+
l
χ0iχ
0j → f f H0
1,2,3, Z f1,2 f1,2
χ+i χ
0j → B+
mB0n H+,W+ χ0
k χ+l
χ+i χ
0j → fufd H+,W+ f ′
d1,2f ′u1,2
χ+i χ
−j → B0
mB0n H0
1,2,3, Z χ+k χ+
l
χ+i χ
−j → B+
mB−n H0
1,2,3, Z, γ χ0k
χ+i χ
−j → fufu H0
1,2,3, Z, γ f ′d1,2
χ+i χ
−j → fdfd H0
1,2,3, Z, γ f ′u1,2
χ+i χ
+j → B+
mB+n χ0
k χ0l
fiχ0j → B0f f f1,2 χ0
l
fdiχ0j → B−fu fd fu1,2 χ+
l
fuiχ0j → B+fd fu fd1,2 χ+
l
fdiχ+j → B0fu fu fd1,2 χ+
l
fuiχ+j → B+fu fd1,2 χ0
l
fdiχ+j → B+fd fu χ0
l
fuiχ−j → B0fd fd fu1,2 χ+
l
fuiχ−j → B−fu fd χ0
l
fdiχ−j → B−fd fu1,2 χ0
l
fdi f∗dj
→ B0mB
0n H0
1,2,3, Z, g fd1,2 fd1,2 p
fdi f∗dj
→ B−mB
+n H0
1,2,3, Z, γ fu1,2 p
fdi f′∗dj
→ f ′′d f
′′′d H0
1,2,3, Z, γ, g χ0k, g
fdi f′∗dj
→ f ′′u f
′′′u H0
1,2,3, Z, γ, g χ+k
fdi f′dj
→ fdf′d χ0
k, g χ0l , g
fui f∗dj
→ B+mB
0n H+,W+ fd1,2 fu1,2 p
fui f′∗dj
→ f ′′u f
′′′d H+,W+ χ0
k, g
fui f′dj
→ f ′′uf
′′′d χ0
k, g χ+l
Table 3: Included coannihilation processes through s−, t−, u−channels and four-point in-
teractions (p). For the fdi f(∗)dj
processes the corresponding process for up-type sfermions canbe obtained by interchanging the u and d indices.
28
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